Correlation Inequalities for Partially Ordered Algebras

The proof of many an inequality in real analysis reduces to the observation that the square of any real number is positive. For example, the AM-GM inequality $$\frac{1}{2}(a + b) \ge \sqrt {ab} $$ is a restatement of the fact that $$(\sqrt {a - \sqrt b } )^2 \ge 0.$$ On the other hand, there exist useful notions of positivity in rings and algebras, for which this ‘positive squares’ property does not hold, viz. the square of an element is not necessarily positive. An interesting example is provided by the polynomial algebra R [x], where one decrees a polynomial to be positive if all its coefficients are positive. A noncommutative example is furnished by the algebra of n × n matrices, where one declares a matrix to be positive if all its entries are positive. Neither example satisfies the positive squares property, however in each case the product of two positive elements is positive.